Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 2 / a / 1 / Solution

This statement is true. If $(x_n)$ is dense in $B_X$, then on $B_{X^{\ast }}$ the weak-star topology is induced by a metric such asThe Banach-Alaoglu theorem makes this metric space compact, and every compact metric space is sequentially compact.